Statistics 270 - Lecture 2
• Last class began Chapter 1 (Section 1.1)
• Introduced main types of data: Quantitative and Qualitative (or
Categorical)
• Discussed ways to describe qualitative data with plots: Bar Chart &
Pie Chart
• Which is more flexible - Pie Chart or Bar Chart? Why?
• Is there a difference in shape of a Bar Chart if show counts or
percentages?
Important Stuff
•
Statistics and Actuarial Science Stats Lab (Statistics Workshop)
• What is Stats Lab for?: One-on-one help is available during its
operation hours.
• Where is it? The Stats Lab is located in K9516 (inside k9510).
How does the Stats Lab Work?
• Statistics Lab Schedule
• The Statistics Workshop opens for regular use from the second week of
classes. The hours will depend on the amount of T.A. time available
and will be posted at the end of the first week of classes. The
Workshop will be open only when there is a T.A. on duty.
• Typically, Mon-Fri: 9:30-16:30
Important Stuff
• Course web page can be found: www.stat.sfu.ca/~dbingham
• Download lecture notes day before class
Important Stuff
•
Assignments: 5-7 of them
•
Will be due Friday, before 4:30 in boxes outside lab
•
The boxes are labelled by course and also by the first letter of your
last name
•
Note:
1. Late assignments will not be accepted
2. Assignments placed in the wrong box (e.g., stat 201) will not
be accepted
3. Class email list…watch out
Important Stuff
•
Office hours: M-W 3:00-4:00…or appointment…
•
Mid-Term: Friday, February 17 ….Day before reading break.
• Some suggested problems:
• Chapter 1: 1, 5, 13 or 14, 19, 26, 29, 33
• Today:
• Finish with Chapter 1.2, 1.3 and start 1.4
• Will do all of Chapter 1…except stem-leaf & time plots
• Please read Chapter 1…nice introduction
Plots for Quantitative Variables
• Can summarize quantitative data using plots
• Most common plots - histogram and box-plots
• Will introduce box-plots later
• Must first distinguish between types of quantitative data
Types of Quantitative Data
• Discrete Variable: A variable is discrete if the set of possible
values is finite or else is countably infinite
• Continuous Variable: A variable is continuous if its possible
values consist of interval(s) on the real line
Example (discrete data)
• In a study of productivity, a large nuber of authors were classified
according to the number of articles they published during a
particular period of time.
Number of
Frequency
articles
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
784
204
127
50
33
28
19
19
6
7
6
7
4
4
5
3
3
Example (continuous data)
• Experiment was conducted to investigate the muzzle velocity of a
anti-personnel weapon (King, 1992)
• Sample of size 16 was taken and the muzzle velocity (MPH)
recorded
279.9
294.1
260.7
277.9
285.5
298.6
265.1
305.3
264.9
262.9
256.0
255.3
276.3
278.2
283.6
250.0
Histogram – Discrete Data
C
o
u
n
t
s
0 2 4 6 8
• Uses rectangles to show
number (or percentage) of
values in intervals
• Y-axis usually displays counts
or percentages
• X-Axis usually shows intervals
• Rectangles are all the same
width
6
8
10
12
14
16
Data
Constructing a Histogram – Discrete Data
• Determine frequency of each value (e.g., number of
articles)
• Mark the possible values for the distribution on an axis
(usually the x-axis)
• Draw a rectangle over each value with height is the
frequency (or relative frequency)
400
200
0
Count
600
800
Example (discrete data)
1
2
3
4
5
6
7
8
9
10
11
Number of Articles
12
13
14
15
16
17
Histogram – continuous data
C
o
u
n
t
s
0 2 4 6 8
• Uses rectangles to show
number (or percentage) of
values in intervals
• Y-axis usually displays counts
or percentages
• X-Axis usually shows intervals
• Rectangles are all the same
width
6
8
10
12
14
16
Data
Constructing a Histogram – continuous data
• Find minimum and maximum values of the data
• Divide range of data into non-overlapping intervals of equal length
• Count number of observations in each interval
• Height of rectangle is number (or percentage) of observations
falling in the interval
• How many categories?
Example
• Experiment was conducted to investigate the muzzle velocity of a
anti-personnel weapon (King, 1992)
• Sample of size 16 was taken and the muzzle velocity (MPH)
recorded
279.9
294.1
260.7
277.9
285.5
298.6
265.1
305.3
264.9
262.9
256.0
255.3
276.3
278.2
283.6
250.0
• What are the minimum and maximum values?
• How do we divide up the range of data?
• What happens if have too many intervals?
• Too Few intervals?
• Suppose have intervals from 240-250 and 250-260. In which
interval is the data point 250 included?
Intervals
[240,250)
[250,260)
[260,270)
[270,280)
[280,290)
[290,300)
[300,310)
Frequency
F
r
e
q
u
n
c
y
0 1 2 3 4
Histogram of Muzzle Velocity
24 0
26 0
28 0
30 0
32 0
Muz
z
le
Velo
What Does a Histogram Demonstrate?
Numerical Summaries
• Graphic procedures visually describe data
• Numerical summaries can quickly capture main features
• Will consider data coming from large populations
Measures of Center
• Have sample of size n from some population, x , x ,..., x
1
2
• An important feature of a sample is its central value.
• Most common measures of center - Mean & Median
n
Sample Mean
• The sample mean is the average of a set of measurements
• The sample mean:
n
x

x  i1n i
Sample Median
• Have a set of n measurements, x , x ,..., x
1
2
n
• Median is point in the data that divides the data in half
• Viewed as the mid-point of the data
• To compute the median:
• Sort the data from smallest to largest
• If n is odd, the median is the
 n 1 


 2 
th
ordered value
• If n is even, the median is the average of the
ordered value
n
 
 2
th
and the
n 
 1
2 
th
Muzzle Velocity Example
Sample Mean vs. Sample Median
• Sometimes sample median is better measure of center
• Sample median less sensitive to unusually large or small values
called
• For symmetric distributions the relative location of the sample mean
and median is
• For skewed distributions the relative locations are
Other Measures of interest
• Maximum
• Minimum
• Percentile - The 100pth percentile is point where at least 100p%
of data are at or above and 100(1-p)% are at or below.
• To compute 100pth percentile of data x , x ,..., x ,
1
2
n
• Order data from largest to smallest
• Compute np
• If np is not an integer, round up to nearest integer, k. The kth ordered
value is the desired percentile.
• If np is an integer, k, the desired percentile is the average of the kth
and (k+1)th ordered values.
Important Percentiles
• First Quartile
• Second Quartile
• Third Quartile
Example
Measures of Spread
• Location of center does not tell whole story
• Usually report measure of center and also spread
• Spread may be different for two distributions, even if center is the
same
Example:
• Lifetime of 2 types of car (100 cars of each brand) ... which would
you buy?
BRAND A
Class (Years)
[6,7)
[7,8)
[8,9)
[9,10)
[10,11)
[11,12)
Frequency
10
15
25
25
15
10
BRAND B
Class (Years)
[6,7)
[7,8)
[8,9)
[9,10)
[10,11)
[11,12)
Frequency
2
8
40
40
8
2
•
Range=Max-Min
•
Interquartile Range (IQR) = Q3-Q1
•
IQR typically better than range because
•
IQR can be used to detect outliers: